SOLUTION: find the least 5 digit number which is exactly divisible by 20,25,30? and find the least number of five digits that is exactly divisible by 16,18,24,and30?

Question 752482: find the least 5 digit number which is exactly divisible by 20,25,30?
and
find the least number of five digits that is exactly divisible by 16,18,24,and30?
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
find the least 5 digit number which is exactly divisible by 20,25,30?

20 = 2*10 = 2*2*5
25 = 5*5
30 = 2*15 = 2*3*5

First we decide the least number of 2's we must have:
a.  20 contains two 2's
b.  25 contains no 2's
c.  30 contains one 2

So to contain 20 as a factor, our number must contain two 2's as factors.

Next we decide the least number of 3's we must have:
a.  20 contains no 3's
b.  25 contains no 3's
c.  30 contains one 3

So to contain 30 as a factor, our number must contain one 3 as a factor

First we decide the least number of 5's we must have:
a.  20 contains one 5
b.  25 contains two 5's
c.  30 contains one 5

So to contain 20 as a factor, our number must contain two 2's as factors

The smallest integer to contain all those as factors is 2*2*3*5*5 = 300.

Trouble is, 300 is only a 3-digit number, and we want a 5-digit number.
so we must multiply 300 by the smallest integer that will make it be at
least 10000, which is the smallest 5-digit number.

We divide 10000 by 300 and get 33.33333...

So the smallest integer greater than that is 34.

So our number = 300*34 or 10200.

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find the least number of five digits that is exactly divisible by 16,18,24,and 30?

16 = 2*8 = 2*2*4 = 2*2*2*2
18 = 2*9 = 2*3*3
24 = 2*12 = 2*2*6 = 2*2*2*3
30 = 2*15 = 2*3*5

First we decide the least number of 2's we must have:
a.  16 contains four 2's
b.  18 contains one 2
c.  24 contains three 2's
d.  30 contains one 2.  

So to contain 16 as a factor, our number must contain four 2's as factors.

Next we decide the least number of 3's we must have:
a.  16 contains no 3's
b.  18 contains two 3's
c.  24 contains one 3
d.  30 contains one 3.

So to contain 18 as a factor, our number must contain two 3's as factors

First we decide the least number of 5's we must have:
a.  16 contains no 5's
b.  18 contains no 5's
c.  24 contains no 5's
d.  30 contains one 5.

So to contain 30 as a factor, our number must contain one 5 as a factor.

The smallest integer to contain all those as factors is 2*2*2*2*3*3*5 = 720.

Trouble is, 720 is only a 3-digit number, and we want a 5-digit number.
so we must multiply 720 by the smallest integer that will make it be at
least 10000, which is the smallest 5-digit number.

We divide 10000 by 720 and get 13.88888...

So the smallest integer greater than that is 14.

So our number = 720*44 or 10080.

Edwin

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